[PPT] - cours ARO07MSSD #5 Random Models of Dynamical Systems Introduction PowerPoint Presentation

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Connection with PDEs Diffusion approximation

cours ARO07–MSSD #5 Random Models of Dynamical Systems Introduction to SDE’s Connection with partial differential equations

Fran¸ cois Le Gland INRIA Rennes + IRMAR people.rennes.inria.fr/Francois.Le_Gland/insa-rennes/ 2 December 2020, via Zoom

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Connection with PDEs Diffusion approximation

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Connection with PDEs Diffusion approximation

this connection between ‚ a second–order partial differential equation (PDE) ‚ and a stochastic differential equation (SDE) works both ways ‚ provides a probabilistic representation for the solution of a PDE, in terms of the solution of a SDE, and makes it possible to design numerical Monte Carlo approximation schemes relying of this probabilistic representation ‚ provides a PDE satisfied by statistics (probability distribution, probability of some event, moment, Laplace transform, etc.) of the solution of a SDE generalizes the method of characteristics, a connection between ‚ a first–order partial differential equation (PDE) ‚ and an ordinary differential equation (ODE)

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Connection with PDEs Diffusion approximation

Introduction: method of characteristics

consider the ordinary differential equation 9 Xptq “ bpXptqq with time–independent coefficient: ‚ a d–dimensional drift vector bpxq defined on Rd it is assumed that the global Lipschitz and linear growth conditions hold associated with this ODE is the first–order partial differential operator M “

d

ÿ

i“1

bip¨q B Bxi such that M f pxq “ f 1pxq bpxq

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Connection with PDEs Diffusion approximation

let upt, xq be the unique (and ’regular enough’) solution to the PDE Bu Bt pt, xq ` M upt, xq “ 0 for any pt, xq in r0, Tq ˆ Rd upT, xq “ φpxq for any x in Rd Theorem 1 upt, xq “ φpX t,xpTqq where X t,xpsq denote the solution at time t ď s ď T of the ODE starting from x P Rd at time t

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Connection with PDEs Diffusion approximation

Proof the chain rule yields d dt ups, X t,xpsqq “ Bu Bt ps, X t,xpsqq ` u1ps, X t,xpsqq bpX t,xpsqq “ Bu Bt ps, X t,xpsqq ` M ups, X t,xpsqq “ 0 since Bu Bt ps, yq ` M ups, yq “ 0 for any y P Rd, and the identity holds in particular for y “ X t,xpsq therefore, the mapping s ÞÑ ups, X t,xpsqq is constant, and in particular its value for s “ t is the same as its value for s “ T, hence upt, xq “ upt, X t,xptqq “ upT, X t,xpTqq “ φpX t,xpTqq l

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Connection with PDEs Diffusion approximation

consider the equation Xptq “ Xp0q ` ż t bpXpsqq ds ` ż t σpXpsqq dBpsq with a m–dimensional Brownian motion B “ pBptq , t ě 0q, and time–independent coefficients: ‚ a d–dimensional drift vector bpxq defined on Rd ‚ a d ˆ m diffusion matrix σpxq defined on Rd it is assumed that the global Lipschitz and linear growth conditions hold associated with this SDE is the second–order partial differential operator L “

d

ÿ

i“1

bip¨q B Bxi ` 1

2 d

ÿ

i,j“1

ai,jp¨q B2 Bxi Bxj it is also assumed that the d ˆ d symmetric matrix apxq “ σpxq σ˚pxq satisfies the uniform ellipticity condition: there exist a positive constant µ ą 0 such that, for any vector ξ P Rd and any point x P Rd it holds

d

ÿ

i,j“1

ai,jpxq ξi ξj “ ξ˚ apxq ξ ě µ|ξ|2

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Connection with PDEs Diffusion approximation

Cauchy initial–value problem

let upt, xq be the unique (and ’regular enough’) solution of the PDE Bu Bt pt, xq ` L upt, xq ´ cpt, xq upt, xq “ f pt, xq for any pt, xq in r0, Tq ˆ Rd upT, xq “ φpxq for any x in Rd Theorem 2 upt, xq “ Et,xrφpXpTqq expt´ ż T

t

cps, Xpsqq dsus ´ Et,xr ż T

t

f ps, Xpsqq expt´ ż s

t

cpr, Xprqq dru dss

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Connection with PDEs Diffusion approximation

Remark introducing the solution upt, xq of the PDE Bu Bt pt, xq ` L upt, xq “ 0 for any pt, xq in r0, Tq ˆ Rd upT, xq “ φpxq for any x in Rd yields Markov semigroup upt, xq “ Et,xrφpXpTqqs “ pQpT ´ tq φqpxq Remark introducing the solution upt, xq of the PDE Bu Bt pt, xq ` L upt, xq ´ λ cpt, xq upt, xq “ 0 for any pt, xq in r0, Tq ˆ Rd upT, xq “ 1 for any x in Rd yields Laplace transform upt, xq “ Et,xrexpt´λ ż T

t

cps, Xpsqq dsus

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Connection with PDEs Diffusion approximation

Remark [duality] recall that xµptq, f y “ xµp0q, f y ` ż t xµpsq, L f y ds

r in some sense

B Bt µptq “ µptq L the mapping t ÞÑ xµptq, uptqy “ ż

E

µpt, dxq upt, xq is constant, i.e. does not depend on the time variable 0 ď t ď T

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Connection with PDEs Diffusion approximation

PDE interpretation d dt xµptq, uptqy “ x B Bt µptq, uptqy ` xµptq, B Bt uptqy “ xµptq L, uptqy ` xµptq, ´L uptqy “ 0 probabilistic interpretation ErφpXpTqqs “ Er ErφpXpTqq | Fptqs s “ Er ErφpXpTqq | Xptqs s “ ż

E

µpt, dxq ErφpXpTqq | Xptq “ xs “ ż

E

µpt, dxq upt, xq “ xµptq, uptqy

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Connection with PDEs Diffusion approximation

Proof (let 0 ď t ď T be fixed throughout the proof) introducing the process V psq “ expt´ ż s

t

cpr, Xprqq dru the usual chain rule yields d dt V psq “ ´cps, Xpsqq V psq

r in integrated form

V psq “ 1 ´ ż s

t

cpr, Xprqq V prq dr since the solution u is ’regular enough’, the Itˆ

formula yields

ups, Xpsqq “ upt, Xptqq ` ż s

t

rBu Bt pr, Xprqq ` L upr, Xprqqs dr ` ż s

t

u1pr, Xprqq σpXprqq dBprq

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Connection with PDEs Diffusion approximation

the integration by parts formula for the process ups, Xpsqq V psq yields ups, Xpsqq V psq “ upt, Xptqq ` ż s

t

rBu Bt pr, Xprqq ` L upr, Xprqqs V prq dr ` ż s

t

V prq u1pr, Xprqq σpXprqq dBprq ` ż s

t

upr, Xprqq r´cpr, Xprqq V prqs dr collecting all the ordinary integral terms reduces to ż s

t

rBu Bt pr, Xprqq ` L upr, Xprqq ´ cpr, Xprqq upr, Xprqqs V prq dr “ ż t

s

f pr, Xprqq V prq dr since Bu Bt pr, yq ` L upr, yq ´ cpr, yq upr, yq “ f pr, yq for any y P Rd, and the identity holds in particular for y “ Xprq

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Connection with PDEs Diffusion approximation

therefore ups, Xpsqq V psq “ upt, Xptqq ` ż s

t

f pr, Xprqq V prq dr ` ż s

t

V prq u1pr, Xprqq σpXprqq dBprq taking s “ T and taking expectation (assuming the integrand belongs to M2pr0, Tsq so that the stochastic integral has zero expectation) yields upt, xq ` Et,xr ż T

t

f pr, Xprqq V prq drs “ Et,xrupT, XpTqq V pTqs “ Et,xrφpXpTqq V pTqs l

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Connection with PDEs Diffusion approximation

Dirichlet boundary–value / initial–boundary–value problems

let D be an bounded open connected subset of Rd, with smooth boundary BD #1 let upxq be the unique (and ’regular enough’) solution of the PDE L upxq ´ cpxq upxq “ f pxq for any x in D upxq “ φpxq for any x on BD Theorem 3 introducing the stopping time τ “ inftt ě 0 : Xptq R Du if such time exists, and τ “ 8 otherwise upxq “ E0,xrφpXpτqq expt´ ż τ cpXpsqq dsus ´ E0,xr ż τ f pXptqq expt´ ż t cpXpsqq dsu dts

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Connection with PDEs Diffusion approximation

Remark introducing the solution upxq of the PDE L upxq “ ´1 for any x in D upxq “ 0 for any x on BD yields mean exit time upxq “ E0,xrτs Remark introducing the solution upxq of the PDE L upxq ´ λ upxq “ 0 for any x in D upxq “ 1 for any x on BD yields Laplace transform of exit time upxq “ E0,xrexpt´λ τus

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Connection with PDEs Diffusion approximation

Remark introducing the solution upxq of the PDE L upxq “ 0 for any x in D upxq “ φpxq for any x on BD yields moment of exit position upxq “ E0,xrφpXpτqqs

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Connection with PDEs Diffusion approximation

the challenge is to apply the Itˆ

formula to a function (the solution u of

the PDE) that is defined only in D, and whose regularity at the boundary BD is not guaranteed introducing ‚ the open ε–interior subset Dε “ tx P D : dpx, BDq ą εu of D ‚ a ’regular enough’ proxy function wε defined on Rd and which coincides on Dε with the solution u of the PDE ‚ the hitting time τε “ inftt ě 0 : Xptq R Dεu if such time exists, and τε “ `8 otherwise it is then possible to apply the Itˆ

formula to the ’regular enough’ proxy

function wε, until the stopping time τε

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Connection with PDEs Diffusion approximation

Proof (sketch of) introducing the process V ptq “ expt´ ż t cpXpsqq dsu the usual chain rule yields d dt V ptq “ ´cpXptqq V ptq

r in integrated form

V ptq “ 1 ´ ż t cpXpsqq V psq ds since the proxy function wε is ’regular enough’, the Itˆ

formula yields

wεpXptqq “ wεpXp0qq ` ż t L wεpXpsqq ds ` ż t w 1

εpXpsqq σpXpsqq dBpsq

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Connection with PDEs Diffusion approximation

the integration by parts formula for the process wεpXptqq V ptq yields wεpXptqq V ptq “ wεpXp0qq ` ż t rL wεpXpsqq ´ cpXpsqq wεpXpsqqs V psq ds ` ż t V psq w 1

εpXpsqq σpXpsqq dBpsq

taking t “ τε ^ T and taking expectation (assuming that the integrand belongs to M2pr0, Tsq so that the stochastic integral has zero expectation, and using the optional sampling theorem for the bounded stopping time τε ^ T) yields E0,xrwεpXpτε ^ Tqq V pτε ^ Tqs “ wεpxq ` E0,xr ż τε^T rL wεpXpsqq ´ cpXpsqq wεpXpsqqs V psq dss where x P Dε

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Connection with PDEs Diffusion approximation

note that Xpsq P Dε and wεpXpsqq “ upXpsqq for any 0 ď s ď τε ^ T, hence E0,xrupXpτε ^ Tqq V pτε ^ Tqs “ upxq ` E0,xr ż τε^T rL upXpsqq ´ cpXpsqq upXpsqqs V psq dss “ upxq ` E0,xr ż τε^T f pXpsqq V psq dss since L upyq ´ cpyq upyq “ f pyq for any y in D, and the identity holds in particular for y “ Xpsq taking the limit as ε Ó 0 and using the Lebesgue dominated convergence theorem yields E0,xrupXpτ ^ Tqq V pτ ^ Tqs “ upxq ` E0,xr ż τ^T f pXpsqq V psq dss (‹)

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Connection with PDEs Diffusion approximation

indeed τε Ò τ almost surely as ε Ó 0, and | ż τε^T f pXpsqq V psq ds| ď ż τε^T |f pXpsqq| ds ď T max

xPD |f pxq|

next, there exists a Lyapunov function h such that L hpxq ě min

zPD hpzq ą 0

for any x P D [Hint: consider hpxq “ expt´λ˚xu] then the Itˆ

formula yields

max

zPD hpzq ě E0,xrhpXpτ ^ Tqs ´ hpxq ě min zPD hpzq E0,xrτ ^ Ts

hence E0,xrτ ^ Ts ď max

zPD hpzq

min

zPD hpzq ă 8

taking the limit as T Ò 8 and using the monotone convergence theorem yields E0,xrτs ă 8, hence τ ă 8 almost surely taking the limit as T Ò 8 in (‹) and using the Lebesgue dominated convergence theorem yields E0,xrupXpτqq V pτqs “ upxq ` E0,xr ż τ f pXpsqq V psq dss l

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Connection with PDEs Diffusion approximation

#2 let upt, xq be the unique (and ’regular enough’) solution of the PDE Bu Bt pt, xq ` L upt, xq ´ cpt, xq upt, xq “ f pt, xq for any pt, xq in r0, Tq ˆ D upT, xq “ φpxq for any x on D upt, xq “ gpt, xq for any x on r0, Tq ˆ BD Theorem 4 * introducing the stopping time τ “ inftt ď s ď T : Xpsq R Du if such time exists, and τ “ T otherwise upt, xq “ Et,xrφpXpTqq expt´ ż T

t

cps, Xpsqq dsu 1tτ “ Tus ` Et,xrgpτ, Xpτqq expt´ ż τ

t

cps, Xpsqq dsu 1tτ ă Tus ´ Et,xr ż τ

t

f ps, Xpsqq expt´ ż s

t

cpr, Xprqq dru dss

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Connection with PDEs Diffusion approximation

Remark let upt, xq be the solution of the PDE Bu Bt pt, xq ` L upt, xq “ 0 for any pt, xq in r0, Tq ˆ D upT, xq “ 0 for any x on D upt, xq “ gpt, xq for any x on r0, Tq ˆ BD then upt, xq “ Et,xrgpτ, Xpτqq 1tτ ă Tus and introducing the solution u1pt, xq of the PDE Bu1 Bt pt, xq ` L u1pt, xq “ 0 for any pt, xq in r0, Tq ˆ D u1pT, xq “ 0 for any x on D u1pt, xq “ 1 for any x on r0, Tq ˆ BD yields exit probability and joint conditional moment of exit time–position u1pt, xq “ Pt,xrτ ă Ts and upt, xq u1pt, xq “ Et,xrgpτ, Xpτqq | τ ă Ts

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Connection with PDEs Diffusion approximation

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Connection with PDEs Diffusion approximation

Diffusion processes

Definition a Markov process X taking values in Rd is called a diffusion process iff (i). for any ε ą 0 and for any t ě 0 and x P Rd 1 h Pr |Xpt ` hq ´ Xptq| ą ε | Xptq “ xs Ñ 0 as h Ó 0 (ii). for any t ě 0 and x P Rd there exists a d–dimensional vector bpxq such that 1 h ErXpt ` hq ´ Xptq | Xptq “ xs Ñ bpxq as h Ó 0 (iii). for any t ě 0 and x P Rd there exists a d ˆ d matrix apxq such that 1 h Er pXpt`hq´Xptqq pXpt`hq´Xptqq˚ | Xptq “ xs Ñ apxq as h Ó 0

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Connection with PDEs Diffusion approximation

Remark a sufficient condition for condition (i) to hold is (i’). for some δ ą 0 and for any t ě 0 and x P Rd 1 h Er |Xpt ` hq ´ Xptq|2`δ | Xptq “ xs Ñ 0 as h Ó 0

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Connection with PDEs Diffusion approximation

Theorem 5 the solution of the stochastic differential equation Xptq “ Xp0q ` ż t bpXpsqq ds ` ż t σpXpsqq dBpsq is a diffusion process Proof ErXpt ` hq ´ Xptq | Xptq “ xs “ Et,xr ż t`h

t

bpXpsqq dss note that uniform integrability holds, hence 1 h Et,xr ż t`h

t

bpXpsqq dss “ Et,xr ż 1 bpXpt ` λ hqq dλs Ñ bpxq as h Ó 0

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Connection with PDEs Diffusion approximation

note that the Itˆ

formula (matrix–valued integration by parts) yields

pXpt ` hq ´ Xptqq pXpt ` hq ´ Xptqq˚ “ ż t`h

t

rbpXpsqq ds ` σpXpsqq dBpsqs pXpsq ´ Xptqq˚ ` ż t`h

t

pXpsq ´ Xptqq rbpXpsqq ds ` σpXpsqq dBpsqs˚ ` ż t`h

t

σpXpsqq σ˚pXpsqq ds

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Connection with PDEs Diffusion approximation

taking expectation (higher–order moments of the solution are bounded, hence the stochastic integral has zero expectation) yields Et,xrpXpt ` hq ´ Xptqq pXpt ` hq ´ Xptqq˚ | Xptq “ xs “ Et,xr ż t`h

t

bpXpsqq pXpsq ´ Xptqq˚ dss ` Et,xr ż t`h

t

pXpsq ´ Xptqq b˚pXpsqq dss ` Et,xr ż t`h

t

apXpsqq dss

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Connection with PDEs Diffusion approximation

note that uniform integrability holds, hence 1 h Et,xr ż t`h

t

bpXpsqq pXpsq ´ Xptqq˚ dss “ Et,xr ż 1 bpXpt ` λ hqq pXpt ` λ hq ´ Xptqq˚ dλs Ñ 0 and 1 h Et,xr ż t`h

t

apXpsqq dss “ Et,xr ż 1 apXpt ` λ hqq dλs Ñ apxq as h Ó 0 l

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Diffusion approximation

for any N ě 1, let pX N

k , k ě 0q denote an homogeneous discrete–time

Markov chain taking values in Rd, characterized by ‚ its initial probability distribution µ (independent of N), ‚ and its (time–independent) transition probability kernel πNpx, dx1q “ PrX N

k P dx1 | X N k´1 “ xs

introduce the vector–valued function bNpxq and the matrix–valued function aNpxq, defined for any x P Rd by ErX N

k ´ X N k´1 | X N k´1 “ xs “

ż

Rdpx1 ´ xq πNpx, dx1q “ bNpxq

N and by ErpX N

k ´ X N k´1q pX N k ´ X N k´1q˚ | X N k´1 “ xs

“ ż

Rdpx1 ´ xq px1 ´ xq˚ πNpx, dx1q “ aNpxq

N

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intuitively, if bNpxq and aNpxq have a finite limit as N Ò 8, then the increments pX N

k ´ X N k´1q have a mean vector and a covariance matrix of

rder 1{N both

introduce the polygonal (piecewise linear) line, interpolating points X N

k at

time instants tN

k , i.e. consider the continuous–time process

X Nptq “ X N

k´1

tN

k ´ t

tN

k ´ tN k´1

` X N

k

t ´ tN

k´1

tN

k ´ tN k´1

for any tN

k´1 ď t ď tN k

“ X N

k´1 ` pX N k ´ X N k´1q t ´ tN k´1

tN

k ´ tN k´1

i.e. (using tN

k “ k{N)

X Nptq “ X N

k´1 `pX N k ´X N k´1q pN t ´pk ´1qq

for any k ´ 1 ď N t ď k

r equivalently

X Nptq “ X N

tN tu ` pX N tN tu`1 ´ X N tN tuq pN t ´ tN tuq

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Connection with PDEs Diffusion approximation

Theorem 6 * assume that for any R ě 0 sup

|x|ďR

|bNpxq ´ bpxq| Ñ 0 and sup

|x|ďR

}aNpxq ´ apxq} Ñ 0 as N Ò 8, and assume that the SDE Xptq “ Xp0q ` ż t bpXpsqq ds ` ż t σpXpsqq dBpsq has a unique solution, where ‚ the matrix–valued function σpxq is such that apxq “ σpxq σ˚pxq ‚ the initial condition Xp0q has probability distribution µ ‚ the process B is a d–dimensional Brownian motion then (under some additional technical assumptions) the sequence X N converges in distribution (on Cpr0, Ts, Rdq) to X here, convergence in distribution means that for any bounded and continuous real–valued function F defined on Cpr0, Ts, Rdq ErFpX Nqs Ñ ErFpXqs as N Ò 8

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Connection with PDEs Diffusion approximation

Wright–Fisher model

simplified model for the reproduction of individuals through the transmissions of alleles (alternative types of the same gene) consider here the case of one gene, with two alleles A and a the population size N is assumed finite and constant at each generation: at generation k, each individual inherits the allele pair of its parent, a randomly (uniformly) selected (with replacement) individual present in the population at generation pk ´ 1q issue: what is the frequency of one allele (say A) in the population present at generation k?

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Connection with PDEs Diffusion approximation

let the random variable Y N

k denote the number of allele of type A present

at generation k the probability for an individual at generation k to pick an allele A is equal to the proportion p “ i{N of allele A available at generation pk ´ 1q, hence conditionally on Y N

k´1 “ i, the random variable Y N k

follows a binomial distribution BinpN, i{Nq therefore, the random variable Y N

k takes values in t0, 1, ¨ ¨ ¨ , Nu, and the

sequence pY N

k , k ě 0q forms a Markov chain with transition probability

matrix πN

i,j “ PrY N k “ j | Y N k´1 “ is “

ˆN j ˙ p i N qj p1 ´ i N qN´j for any i, j P t0, 1, ¨ ¨ ¨ , Nu

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Connection with PDEs Diffusion approximation

recall that a BinpN, pq random variable has mean N p and variance N p p1 ´ pq, so that ErY N

k ´ Y N k´1 | Y N k´1 “ is “ N i

N ´ i “ 0 and ErpY N

k ´ Y N k´1q2 | Y N k´1 “ is “ N i

N p1 ´ i N q “ i p1 ´ i N q thinking more in term of frequencies (i.e. proportions) rather than in terms of number of individuals, introduce the normalized random variable X N

k “ Y N k

N , taking values in t0, 1{N, ¨ ¨ ¨ , 1 ´ 1{N, 1u Ă r0, 1s

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Connection with PDEs Diffusion approximation

for any p P t0, 1{N, ¨ ¨ ¨ , 1 ´ 1{N, 1u, there exists some i P t0, 1, ¨ ¨ ¨ , Nu such that p “ i{N, hence ErX N

k ´ X N k´1 | X N k´1 “ ps “ 1

N ErY N

k ´ Y N k´1 | Y N k´1 “ is “ 0

and ErpX N

k ´ X N k´1q2 | X N k´1 “ ps

“ 1 N2 ErpY N

k ´ Y N k´1q2 | Y N k´1 “ is “ 1

N2 i p1 ´ i N q “ 1 N p p1 ´ pq the conditions for the validity of the diffusion approximation are satisfied, with bNppq “ bppq “ 0 and aNppq “ appq “ p p1 ´ pq

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Connection with PDEs Diffusion approximation

therefore, the continuous–time process X Nptq interpolating points X N

k at

time instants tN

k “ k{N, converges in distribution to the solution of the

SDE Xptq “ Xp0q ` ż t a Xpsq p1 ´ Xpsqq dBpsq extending by continuity the definition of the diffusion coefficient outside the interval r0, 1s, i.e. setting σpxq “ $ & % a x p1 ´ xq if 0 ď x ď 1

therwise

there exists a unique solution to this SDE, taking values in r0, 1s

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Connection with PDEs Diffusion approximation

this approximation makes it possible to address problems such as extinction of one allele or the other indeed, if T0 “ inftt ě 0 : Xptq “ 0u and T1 “ inftt ě 0 : Xptq “ 1u denote the extinction time of allele A or a (i.e. the first time when no individual in the population carries the allele A or a) respectively, then ExrT0 ^ T1s “ gpxq where gpxq “ ´2 rx log x ` p1 ´ xq logp1 ´ xqs is the unique solution of the second order differential equation

1 2 x p1 ´ xq g 2pxq “ ´1

for any x P R with boundary conditions gp0q “ gp1q “ 0

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